As a fundamental connection to solid-state and low-temperature physics, recent studies reveal nuclear-polarized phases of hydrogen atoms embedded in solid H\(_2\)  films [1]}, [2]} with remarkably large deviations of the low-temperature nuclear spin polarization of H atoms from the Boltzmann distribution [2]}, [4]}, [5]}. This finding raises a number of interesting and important questions regarding the role of quantum effects in systems of this class, including the feasibility of quantum entanglement of H at thermal equilibrium. Important insights into this question can be found in earlier studies on electron-spin dynamics in two-electron double-quantum-dot systems [6]}, [7]}, which have been shown to offer much promise as prospective qubits and advantageous building blocks for quantum
information technologies [8]}, [9]}, [10]}.
In search of the insights into these questions, we are led to revisit the entanglement capacity of the basic building block of such a hypothetical quantum interface – a single hydrogen atom. As we show below in this paper, the electron and nuclear spin degrees of freedom in a hydrogen atom provide both a physically meaningful frame and a suitable Hilbert space for bipartite entanglement. We quantify this entanglement in terms of a two-qubit concurrence and quantum coherence, revealing a connection of H-atom entanglement to the fundamental physical constants – the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a \(\tau _c\)  \(\approx \)  5.35 \(\mu \) eV threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement above \(\tau _c\) , enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.
